Non-effectively Hyperbolic Characteristics
In this chapter introducing the notion of local and microlocal elementary factorization of p, arising from standard techniques of energy integrals we prove that if p is of spectral type 1 on Σ then p always admits a local elementary factorization. On the other hand if p is of spectral type 2 even micolocal elementary factorization is not always possible. When p is of spectral type 2 near ρ we prove that p admits a “nice” microlocal factorization near ρ, which is also a microlocal elementary factorization at ρ, if the cube of some vector field H S annihilates p on Σ near ρ. This factorization is crucial to deriving energy estimates.
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